Probability Of Picking Green Balls Calculate The Chances
Hey guys! Let's dive into a probability problem that involves picking balls from a bag. This is a classic scenario in probability, and we're going to break it down step by step so you can totally ace it. So, grab your thinking caps, and let’s get started!
Problem Statement
Okay, so here's the deal: We have a bag with 16 identical balls, and 4 of them are green. A boy is picking a ball at random, and after each pick, he puts the ball back in the bag. This is repeated 5 times. We need to figure out the probability that he:
(i) Did not pick a green ball. (ii) Picked a green ball at least three times.
Understanding the Basics
Before we jump into the calculations, let's make sure we understand the basic concepts. Probability is all about figuring out how likely something is to happen. It's calculated as the number of favorable outcomes divided by the total number of possible outcomes.
In this case, the total number of balls in the bag is 16, and the number of green balls is 4. So, the probability of picking a green ball in a single draw is 4/16, which simplifies to 1/4. That also means the probability of not picking a green ball is 1 - 1/4 = 3/4.
Since the boy replaces the ball each time, each pick is an independent event. This means the outcome of one pick doesn't affect the outcome of the others. This is super important because it allows us to use some cool probability rules.
(i) Probability of Not Picking a Green Ball
Breaking Down the Scenario
First, let's tackle the first part of the problem: What's the probability that the boy doesn't pick a green ball in any of the 5 tries?
Since each pick is independent, we can multiply the probabilities of not picking a green ball for each of the 5 picks. Remember, the probability of not picking a green ball in a single try is 3/4.
Calculating the Probability
So, the probability of not picking a green ball in 5 tries is:
(3/4) * (3/4) * (3/4) * (3/4) * (3/4) = (3/4)^5
Let's calculate that:
(3/4)^5 = 243 / 1024 ≈ 0.2373
So, the probability of the boy not picking a green ball in 5 tries is approximately 0.2373, or about 23.73%.
Why This Makes Sense
Think about it: Each time the boy picks a ball, there's a 75% chance he won't pick a green one. If he does this five times in a row, the probability gets smaller each time, which is why we end up with a probability less than 25%.
(ii) Probability of Picking a Green Ball at Least Three Times
Understanding “At Least Three Times”
Now, let's move on to the second part: What's the probability that the boy picks a green ball at least three times? This means he could pick a green ball 3 times, 4 times, or even 5 times. We need to calculate the probability for each of these scenarios and then add them up.
Using the Binomial Probability Formula
This is where the binomial probability formula comes in handy. The binomial probability formula helps us calculate the probability of getting exactly k successes in n independent trials, where each trial has only two possible outcomes (success or failure). The formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials
- k is the number of successes
- p is the probability of success in a single trial
- (n C k) is the number of combinations of n items taken k at a time, also written as "n choose k"
Applying the Formula
In our case:
- n = 5 (number of picks)
- p = 1/4 (probability of picking a green ball)
- 1 - p = 3/4 (probability of not picking a green ball)
- k = 3, 4, or 5 (number of times we pick a green ball)
We need to calculate P(X = 3), P(X = 4), and P(X = 5) and then add them up.
Calculating P(X = 3)
P(X = 3) = (5 C 3) * (1/4)^3 * (3/4)^2
First, let's calculate (5 C 3), which is "5 choose 3". This is the number of ways to choose 3 items from a set of 5, and it's calculated as:
(5 C 3) = 5! / (3! * (5 - 3)!) = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1)) = 10
Now, plug that into the formula:
P(X = 3) = 10 * (1/4)^3 * (3/4)^2 = 10 * (1/64) * (9/16) = 90 / 1024 ≈ 0.0879
Calculating P(X = 4)
P(X = 4) = (5 C 4) * (1/4)^4 * (3/4)^1
(5 C 4) = 5! / (4! * (5 - 4)!) = 5! / (4! * 1!) = (5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * 1) = 5
P(X = 4) = 5 * (1/4)^4 * (3/4) = 5 * (1/256) * (3/4) = 15 / 1024 ≈ 0.0146
Calculating P(X = 5)
P(X = 5) = (5 C 5) * (1/4)^5 * (3/4)^0
(5 C 5) = 5! / (5! * (5 - 5)!) = 5! / (5! * 0!) = 1 (Remember, 0! is defined as 1)
P(X = 5) = 1 * (1/4)^5 * 1 = 1 / 1024 ≈ 0.00098
Adding the Probabilities
Now, we add the probabilities for 3, 4, and 5 green balls:
P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5) ≈ 0.0879 + 0.0146 + 0.00098 ≈ 0.1035
So, the probability of picking a green ball at least three times is approximately 0.1035, or about 10.35%.
Why This Number Is Smaller
It makes sense that this probability is smaller than the probability of not picking any green balls. Picking a green ball at least three times is a less likely event than not picking any green balls at all, given the probabilities involved.
Final Answer
Alright, guys, we've crunched the numbers and here's what we've got:
(i) The probability that the boy did not pick a green ball is approximately 0.2373. (ii) The probability that the boy picked a green ball at least three times is approximately 0.1035.
Key Takeaways
- Independent Events: When events are independent, we can multiply their probabilities.
- Binomial Probability: The binomial formula is super useful for calculating probabilities in situations with a fixed number of trials and two outcomes.
